Suppose $A, B, C$ are $n\times n$ matrices. Let $A = L_1U_1$ and $D = L_2U_2$. Then what is the LU decomposition of $$\begin{bmatrix} A&B\\ 0&D\end{bmatrix}$$
How to find this?
I am able to find $$\begin{bmatrix} A&B\\ 0&D\end{bmatrix} = \begin{bmatrix} L_1&0\\ 0&L_2\end{bmatrix}\times \begin{bmatrix} U_1&X\\ 0&U_2\end{bmatrix}$$but this means $L_1X = B$ which might not be the case. How to look for this?